Rockfish Historical Harvest Estimates

This project seeks to estimate sport fish harvest and releases of rockfish in Alaska waters by improving on the Howard et al. (2020) methods and expand the time series back to 1977 when the statewide harvest survey (SWHS) was first implemented. This is essentially a Bayesian version of the Howard methods that allows for more appropriate and defensible sharing of information between areas, handles missing data in a more appropriate manor, accurately propagates uncertainty throughout the estimation procedure and replaces the Howard decision tree approach to low sample sizes with a hierarchical model. The methods and results for generating harvest estimates are generally consistent between the Bayesian model and the Howard methods. Harvest estimates are consistent with Howard estimates during contemporary times, but may differ based on more appropriate weighting of SWHS and logbook data, including estimating and correcting bias in the SWHS data.

The Bayesian methods depart from the Howard method in how releases are estimated. The Howard methods assume that the species composition of the harvests are equal to the species composition of released fish, which is clearly contraindicated in the logbook data. For instance, logbook data demonstrates that yelloweye have been retained at high levels up until restrictions were enacted in recent years, whereas pelagic rockfish were released in significant numbers in the past with retention increasing in recent years as they have become more prized by anglers. Recent prohibition on retaining yelloweye in Southeast Alaska highlights the shortcomings of the original Howard assumptions as the species composition of the harvest would indicate that no yelloweye were caught and released during the closure.

The Howard method for estimating releases for private anglers also relied on an expansion of the logbook release estimates based on the ratio of private:guided releases of all rockfish in the SWHS. In addition to the faulty assumptions about species composition, this method ignores potential bias in SWHS estimates of harvests and releases or at least assumes that the bias in release and harvests are the same. As demonstrated in Figure 1, the bias in those two quantities appears to be quite different based on the logbook data. The Bayesian model thus attempts to estimate release probabilities based on the logbook data coupled with bias corrected estimates from the SWHS.

Lastly, the Howard methods were only used on data beginning in 1999 with the advent of the logbook program and estimates of harvests and releases prior to that have been based on linear ramps from 1999 back to the perceived start of the fishery. The Bayesian methods allow us to expand the time series back to 1977 when the SWHS was implemented by leveraging regional data trends in species composition and the proportion of caught rockfish harvested by species and/or species complex. Key advantages of the Bayesian approach are highlighted in table 1.

Table 1. Summary of key improvements in reconstructiing sport fish removals of rockfish using the Bayesian model as compared to the Howard et al. (2020) methods.

Issue	Howard	Bayes
Time series	1999 - present	1977 - present
Bias in SWHS	Not explicitly dealt with. Relies on logbook data and ratios of guided/unguided from SWHS data to estimate unguided releases and harvests.	Explicitly estimates bias in SWHS harvest and release estimates based on logbook data.
Species composition of releases	Assumes that species composition of releases is equal to that of the harvest, which is not evident in the logbook data.	Recognizes different release probabilities by species / species assemblage and estimates it from logbook data and bias corrected SWHS data
Sample size limitations	Uses sample size threshholds such that when areas fall below those threshholds values are borrowed from nearby areas.	Uses a hierarchichacal modelling approach that shares information between areas in the same region. Thus all data is used, even with small sample sizes. This is a more sound method that avoids assumptions and uses all of the data.
Error propogation	Error is propogated when variance estimates are available, but there is uncertainty associated with borrowing values from nearby areas, or the assumption of species compositions being identical in harvest and releases, are not dealt with.	By breaking the assumption that species composition is equal between harvests and releases, uncertainty in the release estimates is more reflective of the fishery. Furthermore, the hyerarchichal approach more accurately captures uncertainy within and between areas within a region.

Data

Harvest data was available for 22 commercial fishing management areas in Southcentral and Southeast Alaska. Areas with negligible rockfish harvest were pooled with adjacent areas for analysis. Specifically the Aleutian and Bering areas were pooled into an area labeled BSAI; the IBS and EKYT were pooled into an area labeled EKYKT; the Southeast, Southwest, SAKPEN and Chignik areas were pooled into an area labeled SOKO2PEN and the Westside and Mainland areas were pooled into an area labeled WKMA.

Stateside Harvest Survey (SWHS)

Statewide harvest survey estimates of rockfish catch and harvest are available for 28 years (1996-2023) for all users and for 13 years (2011-2023) for guided anglers (Figure 0). Additionally, there are overall harvest estimates from 1977- 1995 and release estimates from 1990-1995 that required some partitioning to ascribe to current management units. Harvests in unknown areas were apportioned based on harvest proportions in 1996. Variance estimates are not available for pre-1996 data and as such, the maximum observed coefficient of variation (cv) in each commercial fisheries management unit was applied to the pre-1996 values.

Figure 1.- Data sources for estimating rockfish harvests and releases in ADF&G commercial fisheries management units. Note that initial rockfish harvest estimates were not differentiated into species assemblage or species until 1998 when logbooks began differentiating by pelagic and non-pelagic. Logbooks began to collect data on yelloweye beginning in 2006. Port sampling programs to gather data on species composition of harvests began in 1996 in Southcentral and Kodiak and in 2006 in Southeast.

SWHS estimates are believed to be biased to some degree. These modelling efforts aim to estimate and correct for that bias with the assumption that logbook records are a census of guided harvests and releases.

SWHS Rockfish release estimates are inferred from the difference between catch and harvest estimates.

Adam noted that the first 5 years (23 years counting the historical data) in the SWHS data set for PWSO seem unreasonable (close to zero and not corroborated with logbook estimates). Adam recommended setting these harvests to unknown, but current model development has included the data. Once a satisfactory model has been identified we will exam the effects of censoring the PWSO data.

Creel Surveys

NA

Guide Logbooks

Sport fishing guides have been required to report their harvest of rockfish for 26 years (1998-2023). Reported harvest is also available by assemblage (pelagic vs. non-pelagic). Harvest of yelloweye and “other” (non-pelagic, non-yelloweye) rockfish were reported separately beginning in 2006.

Logbooks also record the number of rockfish released for the same categories. However, the reliability of the release data is somewhat questionable as reported releases are generally far lower than that estimated by the SWHS. As such several treatments of the data are considered.

Logbook versus SWHS estimates

Estimates of guided harvests and releases from the SWHS do not align with the census from charter logbooks. Logbook harvest reports are generally considered reliable and are used to assess the bias in SWHS reports. However, there is even greater disparity between release estimates in the two sources and it is debatable whether logbook releases should be treated as a census. The Howard et al. (2020) methods do treat the logbook release data as “true” and thus are considerably less than would be estimated from the SWHS data.

Figure 2.- SWHS harvest (left) and release (right) estimates from guided trips (x-axis) versus repoted harvests from charter logbooks (y-axis).

A note on model development

To evaluate the discrepancy in apparent bias in harvest and release data, several models were explored to estimate releases during model development. One method (\(LB_{fit}\)) considers the logbook release data to be reliable and a second method (\(LB_{cens}\)) treated the logbook release data as estimates of the minimum released, thus giving more weight to SWHS release estimates. A third method (\(LB_{hyb}\)) is a hybrid approach that treats reported releases of yelloweye as reliable but total rockfish and pelagic rockfish releases as minimums. Model development revealed a tension between the total and pelagic logbook releases and the yelloweye logbook releases. This tensions eventually highlighted the different release/retention probabilities between yelloweye and pelagics in the logbook data and prompted the current approach whereby that probability was calculated for the three main species complexes covered in the data: pelagics, yelloweye, and “other”. The methods described here follow the (\(LB_{fit}\)) formulation. Based on model behavior it is unlikely that the (\(LB_{cens}\)) model would work as there would not be enough data to estimate release probabilities. However, it may be worth running the (\(LB_{hyb}\)) approach as a sensitivity test at the very least.

Composition data

Harvest sampling data exists from Gulf of Alaska areas since 1996 and from Southeast Alaska areas since 2006. Port sampling data is comprised of the number of total rockfish, pelagic and non-pelagic rockfish, black rockfish and yelloweye rockfish. In Southeast Alaska, the number of Demersal Shelf Rockfish (DSR, of which yelloweye are one species) and slope rockfish are also recorded.

Process equations

The true harvest \(H_{ay}\) of rockfish for area \(a\) during year \(y\) is assumed to follow a temporal trend defined by a penalized spline:

\[\begin{equation} \textrm{log}(H_{ay})~\sim~\textrm{Normal}(f(a,y), {\sigma_H}) \end{equation}\]

where \(f(a,y)\) in a p-spline basis with 7 components (knots) and a second degree penalty. The variance, \(\sigma_H\), was given a normal prior with a mean and standard deviation of 0.25 and 1, respectively.

Charter and private harvest \(H_{ayu}\) (where u = 1 for charter anglers and u = 2 for private anglers) is a fraction of total annual harvest in each area:

\[\begin{equation} H_{ay1}~=~H_{ay}P_{(user)ay1}\\H_{ay2}~=~H_{ay}(1-P_{(user)ay1}) \end{equation}\]

where \(P_{(user)ay1}\) is the fraction of the annual harvest in each area taken by charter anglers. \(P_{(user)ay1}\) was modeled hierarchically across years as:

\[\begin{equation} P_{(user)ay1}~\sim~\textrm{beta}(\lambda1_a, \lambda2_a) \end{equation}\]

with non-informative priors on both parameters.

Annual black rockfish harvest \(H_{(black)ayu}\) for each area and user group is:

\[\begin{equation} H_{(black)ayu}~=~H_{ayu}P_{(pelagic)ayu}P_{(black|pelagic)ayu} \end{equation}\]

where \(P_{(pelagic)ayu}\) is the fraction of the annual harvest for each area and user group that was pelagic rockfish and \(P_{(black|pelagic)ayu}\) is the fraction of the annual harvest of pelagic rockfish for each area and user group that was black rockfish.

The southeast region also tracks two other non-pelagic rockfish assemblages, demersal shelf rockfish (DSR, which includes yelloweye) and slope rockfish. For the southeast region the harvest of those two assemblages is thus

\[\begin{equation} H_{(DSR)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(DSR|non-pelagic)ayu}\\ H_{(slope)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(slope|non-pelagic)ayu}\\ \end{equation}\]

where \(P_{(DSR|non-pelagic)ayu}\) and \(P_{(slope|non-pelagic)ayu}\) are the fractions of the annual harvest of non-pelagic rockfish for each area and user group that were DSR and slope rockfish, respectively.

Annual yelloweye rockfish harvest \(H_{(yelloweye)ayu}\) for each area and user group is calculated differently for central/Kodiak areas and southeast areas. For central and Kodiak areas yelloweye rockfish harvests are calculated as

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{ayu}(1-P_{(pelagic)ayu})P_{(yelloweye|non-pelagic)ayu} \end{equation}\]

where \(P_{(yellow|non-pelagic)ayu}\) is the fraction of the annual harvest of non-pelagic rockfish for each area and user group that was yelloweye rockfish.

For southeast areas yelloweye harvests are a fraction of the DSR harvests such that

\[\begin{equation} H_{(yelloweye)ayu}~=~H_{(DSR)ayu}P_{(yelloweye|DSR)ayu} \end{equation}\]

The composition parameters \(P_{(comp)ayu}\), were modeled using a logistic curve that would allow hindcasting without extrapolating beyond the limit of observed values such that:

\[\begin{equation} \textrm{logit}(P_{(comp)ayu})~=~\beta0_{(comp)ayu} + \frac{\beta1_{(comp)ayu}}{(1 + exp(\beta2_{(comp)ayu}*(y - \beta3_{(comp)ayu})))} + \beta4_{(comp)ayu}*I(u=private)+re_{(comp)ayu} \end{equation}\]

where the \(\beta\) parameters define the intercept, scaling factor, slope, inflection point and private angler effect, respectively, \(y\) is the year index, \(I(u=private)\) is an index variable which is 1 when the user groups is private and 0 otherwise and \(re_{(comp)ayu}\) is a random effect with a non-informative prior. \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernible change in composition over the observed time period. \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was used for hindcasting.

The true number of released rockfish \(R_{ayu}\) were based on the proportion of the total catch harvested, \(pH_{(comp)ayu}\), by area, year, user group and species grouping. Because release data from the SWHS is for all rockfish and the release data from logbooks is only subdivided into pelagics, yelloweye and “other” (non-pelagic, non-yelloweye), we only estimated \(pH_{(comp)ayu}\) for those categories. Thus, converting \(H_{(comp)ayu}\) to total catches by user group, \(C_{(comp)ayu}\), with \(pH_{(comp)ayu}\) results in estimates of total releases such that

\[\begin{equation} R_{(comp)ayu}~=~ C_{(comp)ayu} - H_{(comp)ayu} ~=~ \frac{H_{(comp)ayu}}{pH_{(comp)ayu}} - H_{(comp)ayu} \end{equation}\]

with total releases equal to the sum of the compositional releases. For non-yelloweye DSR and Slope rockfish assemblages in Southeast Alaska \(R_{(DSR)ayu}\) and \(R_{(slope)ayu}\) were estimated from \(R_{(other)ayu}\) using the species composition data from the harvest, thus assuming that slope and DSR assemblages were caught and released at the same rates.

The proportion harvest parameters for \(pH_{(comp)ayu}\) were modeled using a logistic curve that would allow hindcasting based on trends in the data without extrapolating beyond the range of observed values such that

\[\begin{equation} \textrm{logit}(pH_{(pH)ayuc})~=~\beta0_{(pH)ayu} + \frac{\beta1_{(pH)ayuc}}{(1 + exp(\beta2_{(pH)ayuc}*(y - \beta3_{(pH)ayuc})))} + \beta4_{(pH)ayuc}*I(u=private)+re_{(pH)ayuc} \end{equation}\]

A random effect term allowed estimation during the historical period when data is available, but the curve defined by the above equation determined release estimates between 1977 and 1990. As with the compositional trends, \(\beta\) parameters were modeled hierarchically by region. When \(\beta\) parameters were inestimable as a result of no discernable change in harvest probability over the observed time period, \(\beta1\) (scaling factor) and \(\beta2\) (slope) were fixed to 0 so that the long term mean value was applied.

Release mortality (i.e., the number of released rockfish expected to die) was calculated assuming fixed mortality rates developed in each of the regions. Deep water release (DWR) devices were mandated for charter fleets in 2013 and rates were derived from CITATION. Southeast applies basic rates estimated in these studies while Southcentral and Kodiak rates were derived by using historical depth-of-release data to adjust the rates based on area and user group.

The total number of mortalities by year, area, user and species/species assemblage in numbers was calculated by summing harvests and release mortality such that

\[\begin{equation} M_{(comp)ayu}~=~ H_{(comp)ayu} + m_{R-(comp)ayu} * R_{(comp)ayu} \end{equation}\]

where \(m_{R-(comp)ayu}\) is the release mortality rate by year, area, user and species (Figure XX).

Total removals in biomass were converted using the average weight of fish from port sampling?. A minimum sample size per year of X fish was used as the cutoff for including in the data set. Weights were modeled hierarchically to estimate weights in years when data was missing. The total biomass of removals by year, area, user and species was thus

\[\begin{equation} B_{(comp)ayu}~=~ \overline{wt}_{(comp)ayu} * M_{(comp)ayu} \end{equation}\]

where \(\overline{wt}_{(comp)ayu}\) is the mean weight by species, area, user and year.

Observation equations

SWHS estimates of annual rockfish harvest \(\widehat{SWHS}_H{ay}\) were assumed to index true harvest:

\[\begin{equation} \widehat{SWHS}_H{ay}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay}b_{ay}), \sigma_{SWHSHay}^2\right) \end{equation}\]

where bias in the SWHS harvest estimates \(b_H{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_H{ay}~\sim~\textrm{Normal}(\mu_H{(b)a}, \sigma_H{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

SWHS estimates of guided angler harvest \(\widehat{SWHS}_H{ay1}\) are related to total harvest by:

\[\begin{equation} \widehat{SWHS}_H{ay1}~\sim~\textrm{LogNormal}\left(\textrm{log}(H_{ay1}b_{ay}), \sigma_{SWHS_{ay1}}^2\right) \end{equation}\]

Reported guide logbook harvest \(\widehat{LB}_H{ay}\) is related to true harvest as:

\[\begin{equation} \widehat{LB}_H{ay}~\sim~\textrm{Poisson}(H_{ay1})\\ \widehat{LB}_H{(pelagic)ay}~\sim~\textrm{Poisson}(H_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_H{(yelloweye)ay}~\sim~\textrm{Poisson}(H_{(yelloweye)ay1})\\ \widehat{LB}_H{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(H_{(nonpel,nonye)ay1})\\ \end{equation}\]

Note that for central and Kodiak areas \(H_{(nonpel,nonye)ay1}\) is equal to the total harvest minus pelagic and yelloweye harvests. For southeast areas \(H_{(nonpel,nonye)ay1}\) is equal to the sum of the DSR and slope harvests minus yelloweye harvests.

SWHS estimates of annual rockfish releases \(\widehat{SWHS}_R{ay}\) were assumed to index true releases in a similar fashion and thus modeled similarly. As such, the release data are related to true releases just as harvests were modeled such that:

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{Poisson}(R_{ay1})\\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{Poisson}(R_{ay1}P_{(pelagic)ay1})\\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Because logbook release data is more questionable and demonstrates greater disagreement with SWHS estimates (Figure 1), a second approaches was considered that loosened the assumption that logbook releases were a census. Methods explored to develope \(LB_{hyb}\) and \(LB_{cens}\) models are detailed at the end of this section.

SWHS estimates of guided angler release \(\widehat{SWHS}_R{ay1}\) is modeled the same as harvests.

SWHS release bias was modeled independently of the harvest bias \(b_H{ay}\) such that

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

where bias in the SWHS release estimates \(b_R{ay}\) is modeled hierarchically across years as:

\[\begin{equation} b_R{ay}~\sim~\textrm{Normal}(\mu_R{(b)a}, \sigma_R{(b)a}) \end{equation}\]

with non-informative priors on both parameters.

The number of pelagic rockfish sampled in harvest sampling programs \(x_{(pelagic)ayu}\) follow a binomial distribution:

\[\begin{equation} x_{(pelagic)ayu}~\sim~\textrm{Binomial}(P_{(pelagic)ayu}, N_{ayu}) \end{equation}\]

where \(N_{ayu}\) is the total number of rockfish sampled in area \(a\) during year \(y\) form user group \(u\). The number of black rockfish sampled in harvest sampling programs was thus a proportion of the pelagic harvests

\[\begin{equation} x_{(black)ayu}~\sim~\textrm{Binomial}(P_{(black)ayu}, N_{ayu}^{pel}) \end{equation}\]

Yelloweye rockfish in Southcentral and Kodiak were modeled similarly as a proportion of the total number of non-pelagics such that

\[\begin{equation} x_{(yellow_{R2})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R2})ayu}, N_{ayu}^{nonpel}) \end{equation}\]

Southeast areas have several other non-pelagic groupings such that DSR and slope rockfish are a proportion of non-pelagics

\[\begin{equation} x_{(DSR)ayu}~\sim~\textrm{Binomial}(P_{(DSR)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

and

\[\begin{equation} x_{(slope)ayu}~\sim~\textrm{Binomial}(P_{(slope)ayu}, N_{ayu}^{nonpel}) \end{equation}\]

with yelloweye in southeast a proportion of the DSR harvest

\[\begin{equation} x_{(yellow_{R1})ayu}~\sim~\textrm{Binomial}(P_{(yellow_{R1})ayu}, N_{ayu}^{DSR}). \end{equation}\].

Kodiak has limited port sampling beyond the main harbors but has a robust hydroacoustic survey that is used to quantify black rockfish abundance across the management area and uses stereocameras to derive species compositions of the hydroacoustic data. This data was used as supplementary data to further inform the model to the proportion of pelagic rockfish that are black in Kodiak areas. Angler landings in Kodiak show a higher proportion of black rockfish relative to the hydroacoustic survey and thus the proportion of black rockfish in the hydroacoustic sample related to the true proportion such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ P_{(black|pelagic)ayu} + ae_{au} \end{equation}\].

where \(ae_{au}\) is the angler effect for each area and user group modeled hierarchically around a mean of 0. Predicted \(P_{(black|pelagic)ayu}^{HA}\) assumed a beta distribution such that

\[\begin{equation} P_{(black|pelagic)ayu}^{HA} ~\sim~ beta(\alpha_{HA},\beta_{HA}) \end{equation}\]

where

\[\begin{equation} \alpha_{HA} ~=~ (P_{(black|pelagic)ayu}^{HA})^2 * \frac{1 - P_{(black|pelagic)ayu}^{HA}}{\frac{var_{P_{HA}}-1}{P_{(black|pelagic)ayu}^{HA}}}, \end{equation}\]

\[\begin{equation} \beta_{HA} ~=~ (\alpha_{HA}) * \frac{1}{P_{(black|pelagic)ayu}^{HA} - 1}, \end{equation}\]

\[\begin{equation} var_{P_{HA}} ~=~ (P_{(black|pelagic)ayu}^{HA} * cvP_{(black|pelagic)ayu}^{HA})^2 \end{equation}\]

where \(cvP_{(black|pelagic)ayu}^{HA}\) is the coefficient of variation for the hydroacoustic proportions

\[\begin{equation} cvP_{(black|pelagic)ayu}^{HA} ~=~ \frac{\sqrt{varP_{(black|pelagic)ayu}^{HA}}}{P_{(black|pelagic)ayu}^{HA}} \end{equation}\]

and the variance is approximated using the XXXX method as

\[\begin{equation} varP_{(black|pelagic)ayu}^{HA} ~=~ (\frac{1}{n_{pel}})^2 * varN_{black} + (\frac{n_{black}}{n_{pel}^2}) * varN_{pel} \end{equation}\]

where \(varN_{black}\) and \(varN_{black}\) are the variance of the estimated number of black and pelagic rockfish in the hydroacoustic survey, respectively (CITATION).

The average weight of rockfish by species, user, area and year was modeled hierarchically at several levels within regions such that

\[\begin{equation} wt_{(comp)ayu} ~\sim~ Normal(wt_{(comp)au},\sigma_{wt_{(comp)au}}) ~\sim~ Normal(wt_{(comp)a},\sigma_{wt_{(comp)a}}) ~\sim~ Normal(wt_{(comp)region},\sigma_{wt_{(comp)region}}) \end{equation}\]

where region refers to Kodiak, Southcentral and Southeast. Mean weights and variance were calculated as XXX.

Alternative likelihoods for release estimates

To loosen the assumption that logbook release data are an effective census of true releases I explored models that treated logbook release estimates as a lower bound on the estimate of true releases. In a hybrid approach yelloweye and non-pelagic releases are regarded as a reliable census (given the emphasis and ease of recording these fish) but censors the pelagic and total rockfish release estimates (where censoring implies NA values) such that

\[\begin{equation} \text{censored} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), 1\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \text{censored} \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), 1\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

This model formulation failed such that there was not enough data to inform pelagic releases and the values did not seem valid. A second approach is being explored that fits the censored data using a lognormal distribution centered around the logbook release value, but also with a lower bound equal to the number of recorded releases such that

\[\begin{equation} \widehat{LB}_R{ay}~\sim~\textrm{LogNormal}\left(\log(R_{ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{ay}, \infty\right) \\ \widehat{LB}_R{(pelagic)ay}~\sim~\textrm{LogNormal}\left(\log(R_{(pelagic)ay}), \sigma_{Ray1}^2\right)\text{T}\left(\widehat{LB}_R{(pelagic)ay}, \infty\right) \\ \widehat{LB}_R{(yelloweye)ay}~\sim~\textrm{Poisson}(R_{(yelloweye)ay1})\\ \widehat{LB}_R{(nonpel,nonye)ay}~\sim~\textrm{Poisson}(R_{(nonpel,nonye)ay1})\\ \end{equation}\]

Logbook data is assumed to be a census and as such there is no estimate of uncertainty. As of this writing, several methods are being examined for how to treat \(\sigma_{Ray1}^2\). Models are being run that attempt to allow the model to estimate \(\sigma_{Ray1}^2\) with priors. A simple model applies a uniform prior (0.1,50) to \(\sigma_{Ray1}^2\). A hierarchichal approach based on regions is also being examined whereby \(\sigma_{Ray1}^2\) is lognormally distributed around hyper priors \(\mu_{\sigma_R}\) and \(\sigma_{\sigma_R}\). Initial efforts have applied a uniform prior on \(\mu_{\sigma_R}\) between 1 and 50 and on \(\sigma_{\sigma_R}\) between 0 and 10.

Priors.

Priors range from uninformative to very informative or fixed. Priors for compositional logistic parameters are in Table 2 and proportion harvest logistic parameters are in Table 3. Until I figure out how to make a nice table in Rmarkdown, please refer to the attached spreadsheet and comp and harvp tabs.

Unresolved issues and outstanding questions:

1. Reliability of unguided release estimates: These estimates have the least information feeding them and rely on the bias-corrected SWHS release estimates of all rockfish and the trends in release probability evident in the logbook data. The \(\beta4\) term that estimates the guided/unguided effect was given a very informative prior that tied the release probability of private anglers tightly to that of the charter fleet. The model is then trying to balance the three species complex estimates (pelagic, yelloweye and other) so that they sum to the total unguided releases estimated from the bias corrected SWHS data. For the most part this seems reasonable and appears to work, but there are certain areas where the estimates are “wonky”:

   1. Total rockfish releases more or less align with the total releases estimated with the Howard methods. Presumably, much of the discrepancy results from the substantial bias in release estimates from the SWHS. Interestingly, the logbook data indicates that the SWHS underestimates harvests but overestimates releases by a significant factor (Figure 23 and 24 below).
   2. In general, release estimates of black rockfish are substantially lower than those calculated using the Howard methods. Presumably, much of this derives from the bias correction of the SWHS release estimates.
   3. Yelloweye release estimates also differ considerably from the Howard estimates, but unlike black rockfish are sometimes lower and sometimes higher. Two areas in particular are a little head scratching. Yelloweye releases in the Kodiak Northeast area in particular are significantly lower than for guided anglers with the same pattern evident in Cook Inlet to a lesser extent. Cook Inlet yelloweye numbers are very small, so this is a sample size issue with little consequence. The cause of the Kodiak northeast estimates is not clear to me at this point, but the model estimates the proportion harvested by unguided anglers to be much lower than that of guided anglers, even with the informative prior on \(\beta4\). This must be a product of the bias corrected SWHS release estimates and how the model is partitioning that estimate into the 3 species complexes, but itis a bit a of head scratcher.

2. Proportion guided estimates: There is not much data on this proportion prior to 2011 and it is not modeled with any sort of trend as was done for species composition and harvest proportions. With the exception of Cook Inlet and North Gulf Coast areas, there is little, if any, trend apparent in the data and perhaps this approach is the best available given the data available. However, if there are data sources somewhere that could inform this part of the model they could be incorporated.

3. Prior choices in general need to be vetted. The priors on the logistic curves are fairly informed in an effort to achieve the desired shapes for hindcasting. Ideally, sensitivity testing would occur but the model is very slow to converge. The beta parameters on the logistic curves have required a lot of work on the priors to reach convergence.

4. Proportion harvest estimates for non-pelagic, non-yelloweye in Kodiak WKMA: I need to adjust the prior on the inflection point, \(\beta3\), so that it is forced to occur after 2006. Right now the model is estimating inflection in two Kodiak areas before that point where there is no data to justify a shift. The current inflection is a result of the hierachichal model.

5. Proportion pelagic in PWS and CSEO: The parameters for these particular proportions are very slow to converge. For the CSEO, the estimates of the \(\beta\) parameters are similar to the other Southeast areas, but the mixing is poor over the length of the chains. In this case I think they will ultimately converge with a very long model run and the shape of the curve in the model output looks acceptable. For the two PWS areas the model seems to struggle with the disparate proportional data from the logbook and the port sampling. There is some wandering in the chains of the \(\beta0\) and \(\beta1\) terms and spikiness in the \(\beta2\) terms. I’ve been working on constraining the hyperpriors for PWS \(beta2\). Similar to CSEO, it may just entail a very long model run to reach convergence, but the shape of the curves looks reasonable.

Next steps:

Once the model is finalized, harvest and release numbers need to be converted into biomass removals. This is a two step process where release mortality estimates are applied to the release estimates to estimate the number of released rockfish that do not survive. This is based on studies and will reflect the values that the department has been using with the Howard methods. Region 2 (both Southcentral and Kodiak) have release-at-depth estimates from a number of years that they apply across all years and then calculate mortality rates based on those estiates. Southease does not have release-at-depth data and simply applies an assumed rate based on research.

Once release mortality is calculated average weight data is applied to convert numbers to biomass. The plan is to incorporate all of this into the model to propogate uncertainty into the posteriors. However, the model already takes a long time to run and I may explore a simpler approach using the posteriors from the numbers model to speed up processing.

Results

• Model: LB_fit_3pH

• Model Run: rf_harvest_est_kha_rm_wt_thru2023_3e+06_7kn_2025-05-29 (Naming convention: modelname_thruYEAR_iters_rundate)

• Proportion of parmaters converged (Rhat <= 1.1): 0.9971

• See end of document for summary of unconverged parameters.

Figure X.- Rhat values and proportion of parameters that converged (Rhat < 1.1.)

Estimate comparison

Since previous estimates of rockfish harvest have been produced these first 3 graphs will be used to show how the modeled estimates compare to the estimates produced earlier. For total rockfish the estimates are in general agreement although differences are noted. These estimates should be more reliable because they include both SWHS and guide logbook data, handle variance more appropriately, use hierarchical distributions when data is missing, directly consider observation error and are produced using reproducible research.

Figure 3.- Total rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 3.- Total rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Notes from Adam: When looking at only black rockfish the most significant differences are for the Prince William Sound Inside area. I did not spend a great deal of time tracking this down although it looks like the previous version used bad values for \(P_{(black)ayu}\) for at least unguided anglers. For the moment I would ignore the results for BSIA and SOKO2SAP. I think it is possible to give approximate values for these areas but it will require a little more coding which I have yet to do.

Figure 4.- Black rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

And black rockfish releases…

Figure 5.- Black rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 6.- Yellow rockfish harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 7.- Yellow rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 8.- DSR rockfish (excluding yelloweye) harvests 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 9.- DSR rockfish releases (including yelloweye) 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 11.- Slope rockfish harvests 1996-2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Figure 12.- Slope rockfish releases 1996–2023. Lines and error polygons represent model estimates and points and error bars represent Howard et al estimates.

Total Biomass Removal Estimates

Figure 13.- Black rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 14.- Yellow rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 15.- Pelagic rockfish estimated total removals in lbs in 1996–2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 16.- Non-yelloweye, demersal shelf rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Figure 17.- Slope rockfish estimated total removals in lbs in 1996-2023. Total removals are calculated from the harvests, the estimated release mortality and mean fish weights. Error polygons represent 95% confidence intervals.

Model fit

Logbook residuals

Figure 18.- Residuals from logbook harvests.

SWHS residuals

Figure 19.- Residuals from SWHS harvests.

Figure 20.- Residual of SWHS releases.

Parameter estimates

P(Charter)

These histograms show the posterior distribution of the mean percent of rockfish harvested by the charter fleet.

Figure 21.- Mean percent of harvest by charter anglers.

When considered annually we see the percent of rockfish harvested by the charter fleet follows our data fairly well although the model smooths out the changes and we just do not have much information about this ratio. Prior to 2011 the percent charter is confounded with SWHS bias and should be mostly discounted.

Figure 22.- Annual estimates of the percent of harvest by charter anglers for 16 commerical fishing manamgent areas, 1996-2023.

P(Harvest)

These plots show the fitted logistic line to the proportion of caught rockfish that are harvested. These estimates are used for hindcasting catch estimates based on the harvest data in early years when catch estimates are unavailable.

Figure 23.- Annual proportion of pelagic rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 24.- Annual proportion of yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available.

Figure 25.- Annual proportion of non-pelagic, non-yelloweye rockfish catch that was harvested. Note that pre-1990 estimates are used to estimate catch in these years when catch estimates are not available. Note, that this is not estimated for Southeast areas because non=pelagics are divided between DSR (including yelloweye) and Slope species.

## NULL

## NULL

SWHS bias

Figure 23 shows the mean estimate for SWHS bias in harvests and releases. Cook Inlet, North Gulf Coast and North Southeast Inside all look pretty good while most other areas have substantial bias. Prince William Sound Inside has the largest bias. Bias in release estimates is substantial and whereas the SWHS appears to underestimate harvests, it appears to greatly overestimates releases by a factor of 2 or more in most areas as derived from logbook reported releases.

Figure 28.- Mean SWHS bias for harvests and catches. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.

Our estimates of SWHS harvest bias track observations fairly well when he have guided harvest estimates. The estimates of release bias in the SWHS data track observed patterns to an extent, but appear to smooth these more volatile disagreements with the logbook data. Adam postulated in his initial start on this that some of this could be the result of the estimates of the proportion guided. This value was not modelled with a trend and thus applies a constant estimate when hindcasting. Data on these relationships could greatly improve this model.

Figure 29.- Annual estimates of SWHS bias in harvests and releases for 16 commerical fishing manamgent areas, 1996-2023. Note that a bias < 1 indicates that the SWHS underestimates the true value and bias > 1 indicates the survey overestimates the true value.

P(pelagic)

We model the percentage of pelagic rockfish in the harvest because we have the information for charter anglers (via logbooks) starting in 1998. Other than looking at the model estimates you can use Figure 25 to compare the two data streams for pelagic rockfish harvest. In general they are in agreement with major exceptions in Price William Sound inside, Prince William Sound outside (early in the time series) and South Southeast inside.

Figure 30.- Annual estimates of the percent of the sport harvest that was pelagic rockfish for 16 commerical fishing manamgent areas, 1996-2023.

P(black|pelagic)

Note that in Southeast Alaska we only have composition data starting in 2006. Tania dug up old SE data, but it did not provide any useful data for species apportionment. For the most part, P(black|pelagic) is relatively constant across areas, with the exception of Cook Inlet and NSEI in Southeast AK. It may be worth discussing whether the shifts in those areas is a result of improved or changing species identification rather than actual shift in the species composition of the catch.

Figure 31.- Annual estimates of the percent of the sport harvest of pelagic rockfish that were black rockfish for 16 commerical fishing manamgent areas, 1996-2023. Kodiak panels include data from a hydroacoustic survey and the proportion of pelagic rockfish that are black in those areas (red) and the adjusted proportions based on obseved harvests for charter (blue) and private (cyan) users.

P(yelloweye|non-pelagic / yelloweye|DSR)

Figure 32.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were yelloweye rockfish for 16 commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

P(DSR|non-pelagic)

Figure 33.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were DSR rockfish for 6 Southeast commerical fishing manamgent areas, 1996-2023.

P(slope|non-pelagic)

Figure 34.- Annual estimates of the percent of the sport harvest of non-pelagic rockfish that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023. Note that P(yelloweye) is the the proportion relative to non-pelagics for Central and Kodiak areas but is relative to DSR for Southeast areas.

P(slope|non-pelagic & non-yellowye) For release estimates

Figure 35.- Annual estimates of the percent of the sport non-pelagic, non-yelloweye releases that were slope rockfish for 6 southeast commerical fishing manamgent areas, 1996-2023.

Weight Fits

Figure 36.- Mean weights of black rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 37.- Mean weights of yelloweye rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 38.- Mean weights of non-black, pelagic rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 39.- Mean weights of non-yelloweye, demersal shelf rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

Figure 40.- Mean weights of slope rockfish used to estimate biomass from harvest and release mortality estimates in numbers of fish.

### Summary of unconverged parameters:

Table 1. Summary of unconverged parameters including the number (n) and the average Rhat from the unconverged parameters.

parameter n badRhat_avg beta2_pH 1 1.247092 beta1_pelagic 2 1.246209 beta3_yellow 1 1.173444 beta1_pH 1 1.170738

parameter n badRhat_avg beta0_pelagic 2 1.144187 beta2_yellow 1 1.123290 beta1_yellow 1 1.116246

Table 2. Summary of unconverged parameters by area

	CI	CSEO	PWSI	SSEI	SSEO	WKMA
beta0_pelagic	0	1	0	0	1	0
beta1_pelagic	0	1	0	0	1	0
beta1_pH	0	0	1	0	0	0
beta1_yellow	1	0	0	0	0	0
beta2_pH	0	0	0	0	0	1
beta2_yellow	0	0	0	1	0	0
beta3_yellow	0	1	0	0	0	0

Parameter estimates:

Summary Table of Parameter Estimates

Parameter	mean	sd	Lower_CI	Median	Upper_CI
mu_bc_H[1]	-0.126	0.072	-0.261	-0.130	0.028
mu_bc_H[2]	-0.095	0.046	-0.176	-0.099	0.003
mu_bc_H[3]	-0.433	0.072	-0.565	-0.435	-0.286
mu_bc_H[4]	-0.991	0.190	-1.374	-0.984	-0.628
mu_bc_H[5]	0.925	0.894	-0.126	0.734	3.227
mu_bc_H[6]	-2.161	0.321	-2.754	-2.170	-1.524
mu_bc_H[7]	-0.460	0.110	-0.680	-0.458	-0.247
mu_bc_H[8]	0.251	0.353	-0.341	0.223	1.052
mu_bc_H[9]	-0.295	0.137	-0.562	-0.296	-0.028
mu_bc_H[10]	-0.106	0.070	-0.234	-0.109	0.035
mu_bc_H[11]	-0.124	0.038	-0.196	-0.125	-0.049
mu_bc_H[12]	-0.252	0.107	-0.480	-0.248	-0.051
mu_bc_H[13]	-0.131	0.079	-0.285	-0.131	0.024
mu_bc_H[14]	-0.301	0.094	-0.488	-0.296	-0.123
mu_bc_H[15]	-0.344	0.050	-0.441	-0.343	-0.241
mu_bc_H[16]	-0.269	0.364	-0.909	-0.296	0.558
mu_bc_R[1]	1.300	0.143	1.026	1.303	1.578
mu_bc_R[2]	1.453	0.094	1.265	1.458	1.636
mu_bc_R[3]	1.388	0.144	1.105	1.387	1.668
mu_bc_R[4]	0.916	0.199	0.492	0.927	1.276
mu_bc_R[5]	1.176	0.463	0.228	1.186	2.045
mu_bc_R[6]	-1.595	0.424	-2.457	-1.595	-0.786
mu_bc_R[7]	0.452	0.211	0.027	0.459	0.844
mu_bc_R[8]	0.552	0.194	0.151	0.554	0.912
mu_bc_R[9]	0.339	0.207	-0.111	0.363	0.705
mu_bc_R[10]	1.297	0.136	1.015	1.301	1.554
mu_bc_R[11]	1.033	0.097	0.838	1.031	1.222
mu_bc_R[12]	0.818	0.203	0.411	0.821	1.204
mu_bc_R[13]	1.026	0.102	0.821	1.028	1.220
mu_bc_R[14]	0.891	0.141	0.608	0.894	1.160
mu_bc_R[15]	0.783	0.114	0.556	0.784	1.011
mu_bc_R[16]	1.096	0.125	0.848	1.097	1.335
tau_pH[1]	5.171	0.450	4.339	5.161	6.084
tau_pH[2]	2.031	0.227	1.608	2.020	2.512
tau_pH[3]	2.131	0.213	1.735	2.118	2.581
beta0_pH[1,1]	0.543	0.179	0.183	0.540	0.877
beta0_pH[2,1]	1.363	0.177	0.987	1.370	1.700
beta0_pH[3,1]	1.433	0.197	1.014	1.440	1.780
beta0_pH[4,1]	1.568	0.227	1.063	1.589	1.948
beta0_pH[5,1]	-0.863	0.288	-1.520	-0.839	-0.366
beta0_pH[6,1]	-0.749	0.459	-1.847	-0.672	-0.084
beta0_pH[7,1]	-0.461	0.442	-1.386	-0.447	0.477
beta0_pH[8,1]	-0.670	0.278	-1.284	-0.635	-0.217
beta0_pH[9,1]	-0.655	0.273	-1.251	-0.634	-0.170
beta0_pH[10,1]	0.236	0.199	-0.174	0.239	0.614
beta0_pH[11,1]	-0.074	0.162	-0.406	-0.067	0.235
beta0_pH[12,1]	0.479	0.186	0.110	0.485	0.827
beta0_pH[13,1]	0.001	0.145	-0.285	0.005	0.279
beta0_pH[14,1]	-0.311	0.169	-0.659	-0.306	0.012
beta0_pH[15,1]	-0.030	0.182	-0.394	-0.027	0.324
beta0_pH[16,1]	-0.448	0.353	-1.349	-0.391	0.072
beta0_pH[1,2]	2.851	0.159	2.528	2.853	3.162
beta0_pH[2,2]	2.884	0.132	2.619	2.889	3.140
beta0_pH[3,2]	3.131	0.149	2.849	3.124	3.446
beta0_pH[4,2]	2.947	0.129	2.698	2.945	3.207
beta0_pH[5,2]	4.844	1.402	3.037	4.556	8.418
beta0_pH[6,2]	3.110	0.206	2.702	3.112	3.520
beta0_pH[7,2]	1.845	0.194	1.452	1.848	2.216
beta0_pH[8,2]	2.874	0.173	2.535	2.873	3.217
beta0_pH[9,2]	3.440	0.217	3.040	3.438	3.872
beta0_pH[10,2]	3.751	0.196	3.366	3.751	4.138
beta0_pH[11,2]	-4.851	0.306	-5.434	-4.847	-4.246
beta0_pH[12,2]	-4.776	0.391	-5.601	-4.763	-4.041
beta0_pH[13,2]	-4.572	0.392	-5.319	-4.576	-3.797
beta0_pH[14,2]	-5.586	0.473	-6.560	-5.564	-4.765
beta0_pH[15,2]	-4.286	0.343	-4.953	-4.297	-3.592
beta0_pH[16,2]	-4.854	0.384	-5.635	-4.838	-4.132
beta0_pH[1,3]	-0.119	0.681	-1.616	-0.041	1.030
beta0_pH[2,3]	2.190	0.162	1.869	2.193	2.498
beta0_pH[3,3]	2.527	0.151	2.241	2.525	2.838
beta0_pH[4,3]	2.968	0.162	2.644	2.971	3.274
beta0_pH[5,3]	2.124	1.327	0.374	1.876	5.350
beta0_pH[6,3]	0.989	0.501	-0.203	1.022	1.884
beta0_pH[7,3]	0.624	0.174	0.297	0.624	0.969
beta0_pH[8,3]	0.307	0.187	-0.057	0.306	0.668
beta0_pH[9,3]	-0.629	0.401	-1.610	-0.590	0.035
beta0_pH[10,3]	0.471	0.381	-0.446	0.509	1.081
beta0_pH[11,3]	-0.150	0.337	-0.805	-0.165	0.497
beta0_pH[12,3]	-0.868	0.355	-1.604	-0.838	-0.264
beta0_pH[13,3]	-0.133	0.310	-0.716	-0.136	0.462
beta0_pH[14,3]	-0.271	0.268	-0.782	-0.275	0.242
beta0_pH[15,3]	-0.716	0.268	-1.232	-0.714	-0.207
beta0_pH[16,3]	-0.381	0.294	-0.943	-0.384	0.221
beta1_pH[1,1]	3.091	0.321	2.514	3.073	3.766
beta1_pH[2,1]	2.162	0.273	1.674	2.144	2.725
beta1_pH[3,1]	1.966	0.311	1.426	1.943	2.653
beta1_pH[4,1]	2.392	0.356	1.827	2.347	3.247
beta1_pH[5,1]	2.294	0.358	1.693	2.263	3.094
beta1_pH[6,1]	3.901	1.060	2.388	3.698	6.411
beta1_pH[7,1]	2.515	0.870	0.848	2.490	4.380
beta1_pH[8,1]	4.008	0.961	2.645	3.824	6.383
beta1_pH[9,1]	2.350	0.385	1.720	2.313	3.168
beta1_pH[10,1]	2.388	0.276	1.864	2.379	2.948
beta1_pH[11,1]	3.255	0.205	2.875	3.246	3.690
beta1_pH[12,1]	2.558	0.215	2.145	2.557	2.986
beta1_pH[13,1]	2.972	0.215	2.569	2.971	3.425
beta1_pH[14,1]	3.415	0.219	3.008	3.410	3.851
beta1_pH[15,1]	2.533	0.227	2.088	2.532	2.994
beta1_pH[16,1]	4.057	0.620	3.172	3.944	5.540
beta1_pH[1,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[2,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[3,2]	0.000	0.004	0.000	0.000	0.001
beta1_pH[4,2]	0.000	0.001	0.000	0.000	0.001
beta1_pH[5,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[6,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[7,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[8,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[9,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[10,2]	0.000	0.000	0.000	0.000	0.000
beta1_pH[11,2]	6.691	0.336	6.041	6.687	7.347
beta1_pH[12,2]	6.444	0.455	5.601	6.416	7.406
beta1_pH[13,2]	6.948	0.432	6.131	6.943	7.804
beta1_pH[14,2]	7.229	0.494	6.335	7.207	8.241
beta1_pH[15,2]	6.760	0.375	6.024	6.757	7.492
beta1_pH[16,2]	7.452	0.433	6.614	7.446	8.329
beta1_pH[1,3]	4.645	1.612	2.122	4.469	8.190
beta1_pH[2,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[3,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[4,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[5,3]	4.049	7.540	0.791	2.787	15.466
beta1_pH[6,3]	3.360	4.631	0.468	2.608	11.537
beta1_pH[7,3]	0.000	0.000	0.000	0.000	0.000
beta1_pH[8,3]	2.739	0.346	2.091	2.730	3.444
beta1_pH[9,3]	2.755	0.466	1.973	2.715	3.917
beta1_pH[10,3]	2.907	0.455	2.149	2.865	4.026
beta1_pH[11,3]	2.738	0.396	1.976	2.726	3.555
beta1_pH[12,3]	4.117	0.441	3.307	4.089	5.021
beta1_pH[13,3]	1.723	0.328	1.074	1.725	2.367
beta1_pH[14,3]	2.520	0.343	1.846	2.519	3.197
beta1_pH[15,3]	2.005	0.294	1.420	2.015	2.563
beta1_pH[16,3]	1.787	0.322	1.155	1.783	2.402
beta2_pH[1,1]	0.480	0.125	0.291	0.458	0.785
beta2_pH[2,1]	0.587	0.369	0.256	0.518	1.296
beta2_pH[3,1]	0.651	0.452	0.231	0.551	1.691
beta2_pH[4,1]	0.478	0.233	0.208	0.440	0.934
beta2_pH[5,1]	1.450	0.990	0.247	1.307	3.862
beta2_pH[6,1]	0.181	0.064	0.090	0.171	0.327
beta2_pH[7,1]	0.070	1.306	0.000	0.000	0.253
beta2_pH[8,1]	0.241	0.084	0.129	0.225	0.446
beta2_pH[9,1]	0.427	0.216	0.174	0.391	0.885
beta2_pH[10,1]	0.617	0.273	0.292	0.559	1.309
beta2_pH[11,1]	0.791	0.211	0.483	0.758	1.291
beta2_pH[12,1]	1.352	0.494	0.739	1.249	2.610
beta2_pH[13,1]	0.740	0.227	0.406	0.708	1.288
beta2_pH[14,1]	0.838	0.214	0.534	0.804	1.335
beta2_pH[15,1]	0.812	0.309	0.421	0.757	1.567
beta2_pH[16,1]	0.391	0.179	0.177	0.339	0.856
beta2_pH[1,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[2,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[3,2]	-2.041	1.827	-6.718	-1.599	-0.029
beta2_pH[4,2]	-2.000	1.818	-6.859	-1.536	-0.031
beta2_pH[5,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[6,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[7,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[8,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[9,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[10,2]	0.000	0.000	0.000	0.000	0.000
beta2_pH[11,2]	-9.382	4.299	-20.435	-8.419	-4.034
beta2_pH[12,2]	-7.836	4.966	-19.828	-6.879	-1.049
beta2_pH[13,2]	-7.636	4.874	-19.811	-6.593	-1.675
beta2_pH[14,2]	-8.371	4.578	-19.566	-7.368	-2.472
beta2_pH[15,2]	-9.141	4.288	-19.895	-8.194	-3.722
beta2_pH[16,2]	-9.411	4.253	-20.108	-8.503	-4.008
beta2_pH[1,3]	0.256	0.393	0.101	0.182	0.726
beta2_pH[2,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[3,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[4,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[5,3]	9.040	6.489	-0.458	8.009	24.354
beta2_pH[6,3]	9.178	6.353	0.189	8.167	24.081
beta2_pH[7,3]	0.000	0.000	0.000	0.000	0.000
beta2_pH[8,3]	10.045	5.762	1.798	8.931	24.422
beta2_pH[9,3]	8.939	6.298	0.485	7.790	24.144
beta2_pH[10,3]	8.649	6.607	0.479	7.465	24.138
beta2_pH[11,3]	-2.265	2.081	-7.856	-1.677	-0.613
beta2_pH[12,3]	-2.433	2.022	-8.334	-1.848	-0.931
beta2_pH[13,3]	-2.904	2.484	-9.641	-2.138	-0.778
beta2_pH[14,3]	-2.824	2.267	-9.116	-2.118	-0.957
beta2_pH[15,3]	-2.932	2.290	-9.700	-2.203	-1.007
beta2_pH[16,3]	-2.974	2.523	-10.599	-2.164	-0.910
beta3_pH[1,1]	35.911	0.811	34.409	35.886	37.570
beta3_pH[2,1]	33.555	1.155	31.546	33.468	36.246
beta3_pH[3,1]	33.664	1.050	31.647	33.623	35.852
beta3_pH[4,1]	33.836	1.203	31.686	33.748	36.354
beta3_pH[5,1]	27.665	1.040	26.465	27.454	30.577
beta3_pH[6,1]	38.250	3.126	32.469	38.103	44.785
beta3_pH[7,1]	30.682	8.022	18.608	29.902	45.186
beta3_pH[8,1]	39.904	2.059	36.274	39.711	44.582
beta3_pH[9,1]	30.703	1.534	28.215	30.583	33.997
beta3_pH[10,1]	32.692	0.905	30.999	32.664	34.663
beta3_pH[11,1]	30.354	0.463	29.436	30.346	31.280
beta3_pH[12,1]	30.170	0.403	29.325	30.176	30.938
beta3_pH[13,1]	33.154	0.591	32.012	33.149	34.390
beta3_pH[14,1]	32.029	0.463	31.154	32.023	32.971
beta3_pH[15,1]	31.182	0.655	29.914	31.188	32.431
beta3_pH[16,1]	32.046	1.021	30.244	31.942	34.291
beta3_pH[1,2]	29.654	7.906	18.401	28.499	44.752
beta3_pH[2,2]	29.897	7.900	18.415	29.023	44.912
beta3_pH[3,2]	30.059	8.007	18.398	29.308	44.809
beta3_pH[4,2]	30.024	8.068	18.442	29.094	44.978
beta3_pH[5,2]	29.835	7.953	18.408	28.705	44.971
beta3_pH[6,2]	29.855	7.731	18.504	28.976	44.769
beta3_pH[7,2]	30.050	8.147	18.385	29.001	44.937
beta3_pH[8,2]	30.004	7.996	18.537	28.877	44.965
beta3_pH[9,2]	29.735	7.957	18.445	28.522	44.884
beta3_pH[10,2]	29.716	7.830	18.400	28.673	44.577
beta3_pH[11,2]	43.405	0.176	43.123	43.387	43.762
beta3_pH[12,2]	43.191	0.187	42.921	43.146	43.695
beta3_pH[13,2]	43.866	0.151	43.467	43.905	44.052
beta3_pH[14,2]	43.304	0.204	43.046	43.254	43.805
beta3_pH[15,2]	43.419	0.195	43.110	43.403	43.817
beta3_pH[16,2]	43.491	0.188	43.156	43.485	43.842
beta3_pH[1,3]	39.126	3.201	33.155	38.993	45.300
beta3_pH[2,3]	30.219	7.947	18.483	29.317	44.849
beta3_pH[3,3]	30.180	8.011	18.453	29.259	44.908
beta3_pH[4,3]	30.171	7.928	18.427	29.251	44.729
beta3_pH[5,3]	36.763	3.890	31.193	36.245	45.018
beta3_pH[6,3]	40.349	3.515	31.740	40.749	45.581
beta3_pH[7,3]	38.015	4.248	31.370	37.820	45.510
beta3_pH[8,3]	41.492	0.253	41.068	41.493	41.948
beta3_pH[9,3]	33.479	0.579	31.737	33.560	34.333
beta3_pH[10,3]	35.812	0.809	33.428	36.004	36.875
beta3_pH[11,3]	41.790	0.825	40.131	41.827	43.244
beta3_pH[12,3]	41.728	0.395	40.979	41.737	42.494
beta3_pH[13,3]	42.727	0.843	41.114	42.735	44.509
beta3_pH[14,3]	41.094	0.593	39.859	41.132	42.167
beta3_pH[15,3]	42.597	0.643	41.218	42.668	43.691
beta3_pH[16,3]	42.870	0.774	41.145	42.969	44.173
beta0_pelagic[1]	2.220	0.133	1.959	2.221	2.480
beta0_pelagic[2]	1.507	0.123	1.271	1.509	1.747
beta0_pelagic[3]	-0.345	0.690	-2.013	-0.176	0.589
beta0_pelagic[4]	-0.312	0.839	-2.347	-0.084	0.848
beta0_pelagic[5]	1.198	0.251	0.697	1.205	1.689
beta0_pelagic[6]	1.466	0.277	0.863	1.488	1.972
beta0_pelagic[7]	1.599	0.213	1.197	1.593	2.045
beta0_pelagic[8]	1.762	0.207	1.359	1.757	2.185
beta0_pelagic[9]	2.480	0.316	1.865	2.484	3.047
beta0_pelagic[10]	2.534	0.206	2.092	2.540	2.924
beta0_pelagic[11]	0.257	0.434	-0.910	0.402	0.754
beta0_pelagic[12]	1.686	0.144	1.412	1.685	1.969
beta0_pelagic[13]	0.319	0.202	-0.130	0.332	0.660
beta0_pelagic[14]	-0.078	0.274	-0.675	-0.052	0.386
beta0_pelagic[15]	-0.255	0.142	-0.530	-0.251	0.018
beta0_pelagic[16]	0.349	0.257	-0.356	0.405	0.692
beta1_pelagic[1]	0.000	0.000	0.000	0.000	0.000
beta1_pelagic[2]	0.000	0.000	0.000	0.000	0.000
beta1_pelagic[3]	1.871	1.188	0.449	1.547	5.214
beta1_pelagic[4]	1.670	0.963	0.357	1.423	3.932
beta1_pelagic[5]	-0.083	0.307	-0.694	-0.081	0.524
beta1_pelagic[6]	-0.102	0.459	-0.879	-0.139	0.763
beta1_pelagic[7]	-0.023	0.295	-0.591	-0.025	0.572
beta1_pelagic[8]	0.004	0.286	-0.535	0.002	0.576
beta1_pelagic[9]	0.214	0.490	-0.784	0.326	0.980
beta1_pelagic[10]	0.060	0.267	-0.450	0.051	0.618
beta1_pelagic[11]	3.173	1.002	2.080	2.775	5.838
beta1_pelagic[12]	2.743	0.293	2.179	2.741	3.318
beta1_pelagic[13]	2.829	0.677	1.748	2.766	4.369
beta1_pelagic[14]	4.165	1.010	2.738	3.965	6.531
beta1_pelagic[15]	2.906	0.266	2.393	2.902	3.441
beta1_pelagic[16]	3.414	0.783	2.636	3.193	5.861
beta2_pelagic[1]	0.000	0.000	0.000	0.000	0.000
beta2_pelagic[2]	0.000	0.000	0.000	0.000	0.000
beta2_pelagic[3]	0.841	2.494	0.035	0.170	8.044
beta2_pelagic[4]	1.486	3.291	0.040	0.402	12.147
beta2_pelagic[5]	-0.004	0.647	-1.350	-0.015	1.368
beta2_pelagic[6]	-0.097	0.705	-1.568	-0.129	1.348
beta2_pelagic[7]	-0.004	0.677	-1.378	0.001	1.447
beta2_pelagic[8]	-0.023	0.646	-1.366	-0.013	1.380
beta2_pelagic[9]	0.200	0.686	-1.277	0.263	1.574
beta2_pelagic[10]	0.010	0.623	-1.368	0.012	1.278
beta2_pelagic[11]	3.389	4.914	0.117	1.437	17.480
beta2_pelagic[12]	7.058	5.778	1.317	5.319	22.847
beta2_pelagic[13]	1.228	2.963	0.199	0.492	8.731
beta2_pelagic[14]	0.355	0.275	0.164	0.304	0.832
beta2_pelagic[15]	6.977	5.237	1.370	5.599	21.117
beta2_pelagic[16]	5.968	5.707	0.210	4.639	21.543
beta3_pelagic[1]	29.885	7.900	18.475	28.648	44.832
beta3_pelagic[2]	29.796	7.980	18.501	28.690	44.795
beta3_pelagic[3]	29.724	6.166	19.155	29.193	43.713
beta3_pelagic[4]	25.104	5.175	18.629	24.384	41.261
beta3_pelagic[5]	29.890	8.194	18.489	28.347	45.412
beta3_pelagic[6]	31.499	7.013	18.974	31.208	44.682
beta3_pelagic[7]	29.694	7.979	18.459	28.623	44.989
beta3_pelagic[8]	29.545	8.013	18.369	28.199	44.999
beta3_pelagic[9]	30.949	6.129	19.236	30.923	43.160
beta3_pelagic[10]	29.367	8.219	18.302	27.768	45.173
beta3_pelagic[11]	42.655	1.518	38.426	43.072	45.030
beta3_pelagic[12]	43.453	0.260	43.008	43.444	43.936
beta3_pelagic[13]	42.713	1.249	40.303	42.738	45.329
beta3_pelagic[14]	42.189	1.651	38.914	42.189	45.472
beta3_pelagic[15]	43.200	0.263	42.589	43.195	43.690
beta3_pelagic[16]	43.177	0.665	41.429	43.244	44.263
mu_beta0_pelagic[1]	0.691	1.048	-1.636	0.779	2.719
mu_beta0_pelagic[2]	1.808	0.393	0.951	1.820	2.563
mu_beta0_pelagic[3]	0.381	0.458	-0.604	0.396	1.280
tau_beta0_pelagic[1]	0.514	0.593	0.050	0.315	2.134
tau_beta0_pelagic[2]	2.717	2.917	0.240	1.979	9.250
tau_beta0_pelagic[3]	1.618	1.261	0.182	1.314	4.845
beta0_yellow[1]	-0.541	0.195	-0.980	-0.523	-0.215
beta0_yellow[2]	0.503	0.162	0.173	0.512	0.801
beta0_yellow[3]	-0.330	0.196	-0.760	-0.324	0.027
beta0_yellow[4]	0.844	0.272	0.050	0.892	1.205
beta0_yellow[5]	-0.292	0.353	-0.993	-0.293	0.405
beta0_yellow[6]	1.118	0.165	0.789	1.117	1.433
beta0_yellow[7]	0.980	0.157	0.674	0.981	1.290
beta0_yellow[8]	1.010	0.153	0.711	1.009	1.320
beta0_yellow[9]	0.658	0.159	0.346	0.658	0.979
beta0_yellow[10]	0.578	0.141	0.305	0.578	0.866
beta0_yellow[11]	-1.946	0.456	-2.788	-1.957	-1.023
beta0_yellow[12]	-3.693	0.421	-4.584	-3.667	-2.928
beta0_yellow[13]	-3.715	0.480	-4.757	-3.680	-2.883
beta0_yellow[14]	-2.081	0.588	-3.037	-2.144	-0.384
beta0_yellow[15]	-2.846	0.425	-3.729	-2.826	-2.064
beta0_yellow[16]	-2.367	0.445	-3.168	-2.381	-1.474
beta1_yellow[1]	0.809	1.112	0.010	0.654	2.621
beta1_yellow[2]	1.070	0.362	0.589	1.026	1.814
beta1_yellow[3]	0.715	0.268	0.233	0.700	1.278
beta1_yellow[4]	1.354	0.732	0.645	1.170	3.691
beta1_yellow[5]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[6]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[7]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[8]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[9]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[10]	0.000	0.000	0.000	0.000	0.000
beta1_yellow[11]	2.097	0.454	1.164	2.112	2.955
beta1_yellow[12]	2.487	0.434	1.695	2.465	3.405
beta1_yellow[13]	2.831	0.478	2.014	2.804	3.878
beta1_yellow[14]	2.178	0.563	0.819	2.218	3.178
beta1_yellow[15]	2.094	0.420	1.339	2.071	2.971
beta1_yellow[16]	2.128	0.447	1.204	2.142	2.972
beta2_yellow[1]	-3.403	2.927	-10.552	-2.664	-0.032
beta2_yellow[2]	-3.381	2.888	-10.712	-2.612	-0.206
beta2_yellow[3]	-3.321	2.973	-10.888	-2.395	-0.151
beta2_yellow[4]	-2.697	2.680	-9.570	-1.905	-0.094
beta2_yellow[5]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[6]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[7]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[8]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[9]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[10]	0.000	0.000	0.000	0.000	0.000
beta2_yellow[11]	-4.974	2.947	-12.259	-4.391	-1.179
beta2_yellow[12]	-5.324	2.949	-12.611	-4.673	-1.493
beta2_yellow[13]	-5.146	2.813	-12.478	-4.542	-1.591
beta2_yellow[14]	-5.154	3.124	-12.736	-4.584	-0.298
beta2_yellow[15]	-5.008	3.367	-13.964	-4.113	-1.062
beta2_yellow[16]	-5.384	3.032	-12.791	-4.694	-1.425
beta3_yellow[1]	25.967	7.158	18.228	22.848	43.955
beta3_yellow[2]	29.161	1.799	25.671	28.974	32.761
beta3_yellow[3]	33.079	3.086	26.635	32.936	40.315
beta3_yellow[4]	29.053	3.470	22.000	28.052	35.829
beta3_yellow[5]	29.895	7.919	18.373	29.107	44.946
beta3_yellow[6]	29.978	8.028	18.490	29.200	44.827
beta3_yellow[7]	30.042	7.962	18.349	29.287	44.647
beta3_yellow[8]	29.961	8.005	18.436	29.056	44.829
beta3_yellow[9]	29.522	7.936	18.424	28.171	44.858
beta3_yellow[10]	30.277	7.898	18.557	29.421	44.722
beta3_yellow[11]	45.246	0.882	43.929	45.383	45.973
beta3_yellow[12]	43.312	0.365	42.566	43.288	44.049
beta3_yellow[13]	44.871	0.393	43.999	44.945	45.562
beta3_yellow[14]	43.811	2.174	34.702	44.175	45.843
beta3_yellow[15]	45.149	0.531	44.150	45.113	45.971
beta3_yellow[16]	44.529	0.718	43.373	44.528	45.808
mu_beta0_yellow[1]	0.106	0.556	-1.001	0.110	1.321
mu_beta0_yellow[2]	0.638	0.343	-0.122	0.661	1.267
mu_beta0_yellow[3]	-2.441	0.637	-3.440	-2.530	-0.845
tau_beta0_yellow[1]	1.760	2.114	0.104	1.137	6.964
tau_beta0_yellow[2]	3.458	3.796	0.318	2.366	13.559
tau_beta0_yellow[3]	1.447	2.106	0.095	0.891	5.905
beta0_black[1]	-0.080	0.158	-0.391	-0.083	0.236
beta0_black[2]	1.919	0.130	1.660	1.921	2.170
beta0_black[3]	1.315	0.135	1.051	1.314	1.577
beta0_black[4]	2.432	0.137	2.162	2.433	2.693
beta0_black[5]	4.637	2.082	1.774	4.188	10.053
beta0_black[6]	4.618	1.978	2.246	4.111	9.748
beta0_black[7]	3.746	1.866	1.550	3.285	8.640
beta0_black[8]	0.947	0.211	0.548	0.943	1.359
beta0_black[9]	2.609	0.233	2.170	2.607	3.059
beta0_black[10]	1.456	0.134	1.194	1.457	1.713
beta0_black[11]	3.484	0.151	3.188	3.483	3.774
beta0_black[12]	4.864	0.176	4.521	4.861	5.209
beta0_black[13]	-0.140	0.277	-0.680	-0.124	0.330
beta0_black[14]	2.855	0.157	2.540	2.855	3.163
beta0_black[15]	1.297	0.158	0.983	1.299	1.616
beta0_black[16]	4.274	0.162	3.962	4.275	4.584
beta2_black[1]	7.581	9.496	0.493	3.567	38.160
beta2_black[2]	0.000	0.000	0.000	0.000	0.000
beta2_black[3]	0.000	0.000	0.000	0.000	0.000
beta2_black[4]	0.000	0.000	0.000	0.000	0.000
beta2_black[5]	0.000	0.000	0.000	0.000	0.000
beta2_black[6]	0.000	0.000	0.000	0.000	0.000
beta2_black[7]	0.000	0.000	0.000	0.000	0.000
beta2_black[8]	0.000	0.000	0.000	0.000	0.000
beta2_black[9]	0.000	0.000	0.000	0.000	0.000
beta2_black[10]	0.000	0.000	0.000	0.000	0.000
beta2_black[11]	0.000	0.000	0.000	0.000	0.000
beta2_black[12]	0.000	0.000	0.000	0.000	0.000
beta2_black[13]	-1.758	1.491	-6.075	-1.270	-0.283
beta2_black[14]	0.000	0.000	0.000	0.000	0.000
beta2_black[15]	0.000	0.000	0.000	0.000	0.000
beta2_black[16]	0.000	0.000	0.000	0.000	0.000
beta3_black[1]	41.776	1.258	39.800	41.941	43.377
beta3_black[2]	25.000	0.000	25.000	25.000	25.000
beta3_black[3]	25.000	0.000	25.000	25.000	25.000
beta3_black[4]	25.000	0.000	25.000	25.000	25.000
beta3_black[5]	25.000	0.000	25.000	25.000	25.000
beta3_black[6]	25.000	0.000	25.000	25.000	25.000
beta3_black[7]	25.000	0.000	25.000	25.000	25.000
beta3_black[8]	25.000	0.000	25.000	25.000	25.000
beta3_black[9]	25.000	0.000	25.000	25.000	25.000
beta3_black[10]	25.000	0.000	25.000	25.000	25.000
beta3_black[11]	25.000	0.000	25.000	25.000	25.000
beta3_black[12]	25.000	0.000	25.000	25.000	25.000
beta3_black[13]	39.099	1.195	36.802	39.281	40.602
beta3_black[14]	25.000	0.000	25.000	25.000	25.000
beta3_black[15]	25.000	0.000	25.000	25.000	25.000
beta3_black[16]	25.000	0.000	25.000	25.000	25.000
beta4_black[1]	-0.258	0.195	-0.645	-0.256	0.119
beta4_black[2]	0.242	0.181	-0.113	0.240	0.599
beta4_black[3]	-0.931	0.193	-1.303	-0.929	-0.560
beta4_black[4]	0.422	0.218	0.001	0.422	0.849
beta4_black[5]	0.523	1.316	-1.405	0.290	3.743
beta4_black[6]	0.549	1.314	-1.241	0.332	3.739
beta4_black[7]	0.436	1.228	-1.277	0.252	3.207
beta4_black[8]	-0.239	0.319	-0.877	-0.236	0.388
beta4_black[9]	0.840	0.804	-0.294	0.689	2.735
beta4_black[10]	0.054	0.183	-0.301	0.052	0.411
beta4_black[11]	-0.695	0.213	-1.132	-0.693	-0.282
beta4_black[12]	0.173	0.322	-0.446	0.169	0.815
beta4_black[13]	-1.186	0.223	-1.621	-1.186	-0.757
beta4_black[14]	-0.182	0.235	-0.641	-0.186	0.279
beta4_black[15]	-0.890	0.218	-1.308	-0.892	-0.467
beta4_black[16]	-0.591	0.228	-1.044	-0.591	-0.143
mu_beta0_black[1]	1.283	0.947	-0.845	1.323	3.125
mu_beta0_black[2]	2.670	1.041	0.722	2.588	4.952
mu_beta0_black[3]	2.501	0.977	0.359	2.549	4.266
tau_beta0_black[1]	0.624	0.600	0.056	0.437	2.209
tau_beta0_black[2]	0.429	0.577	0.046	0.247	1.896
tau_beta0_black[3]	0.243	0.168	0.051	0.202	0.652
beta0_dsr[11]	-2.885	0.289	-3.454	-2.880	-2.324
beta0_dsr[12]	4.562	0.283	4.024	4.561	5.134
beta0_dsr[13]	-1.356	0.330	-1.992	-1.344	-0.772
beta0_dsr[14]	-3.663	0.517	-4.665	-3.663	-2.633
beta0_dsr[15]	-1.935	0.286	-2.492	-1.930	-1.388
beta0_dsr[16]	-2.981	0.365	-3.704	-2.972	-2.285
beta1_dsr[11]	4.829	0.301	4.242	4.830	5.421
beta1_dsr[12]	7.565	23.897	2.219	5.051	20.457
beta1_dsr[13]	2.863	0.370	2.240	2.853	3.479
beta1_dsr[14]	6.324	0.545	5.251	6.331	7.396
beta1_dsr[15]	3.337	0.289	2.782	3.340	3.925
beta1_dsr[16]	5.792	0.379	5.064	5.789	6.549
beta2_dsr[11]	-8.063	2.292	-13.333	-7.750	-4.386
beta2_dsr[12]	-6.984	2.714	-13.032	-6.805	-2.175
beta2_dsr[13]	-6.364	2.672	-12.076	-6.228	-1.710
beta2_dsr[14]	-6.054	2.677	-11.833	-5.898	-1.695
beta2_dsr[15]	-7.675	2.406	-13.170	-7.364	-3.852
beta2_dsr[16]	-7.875	2.371	-13.613	-7.532	-4.196
beta3_dsr[11]	43.489	0.148	43.217	43.488	43.772
beta3_dsr[12]	33.956	0.777	32.078	34.116	34.803
beta3_dsr[13]	43.251	0.306	42.777	43.198	43.851
beta3_dsr[14]	43.366	0.247	43.080	43.291	43.958
beta3_dsr[15]	43.503	0.188	43.156	43.503	43.853
beta3_dsr[16]	43.440	0.158	43.170	43.427	43.762
beta4_dsr[11]	0.577	0.219	0.159	0.575	1.020
beta4_dsr[12]	0.247	0.433	-0.602	0.243	1.128
beta4_dsr[13]	-0.162	0.221	-0.603	-0.160	0.276
beta4_dsr[14]	0.151	0.253	-0.364	0.152	0.636
beta4_dsr[15]	0.725	0.220	0.307	0.724	1.152
beta4_dsr[16]	0.154	0.224	-0.299	0.157	0.600
beta0_slope[11]	-1.846	0.147	-2.134	-1.847	-1.551
beta0_slope[12]	-4.474	0.259	-4.998	-4.473	-3.968
beta0_slope[13]	-1.343	0.180	-1.726	-1.331	-1.026
beta0_slope[14]	-2.672	0.167	-3.014	-2.672	-2.349
beta0_slope[15]	-1.339	0.147	-1.622	-1.341	-1.048
beta0_slope[16]	-2.741	0.161	-3.064	-2.741	-2.434
beta1_slope[11]	4.477	0.219	4.065	4.478	4.924
beta1_slope[12]	3.976	0.447	3.124	3.976	4.870
beta1_slope[13]	2.710	0.427	2.193	2.641	3.933
beta1_slope[14]	6.310	0.409	5.530	6.312	7.112
beta1_slope[15]	3.000	0.208	2.593	2.998	3.405
beta1_slope[16]	5.295	0.289	4.745	5.296	5.873
beta2_slope[11]	8.573	2.249	5.097	8.230	13.834
beta2_slope[12]	6.570	2.899	1.228	6.620	12.619
beta2_slope[13]	5.347	3.034	0.437	5.190	11.589
beta2_slope[14]	6.277	2.518	2.208	6.122	11.705
beta2_slope[15]	8.157	2.309	4.521	7.841	13.472
beta2_slope[16]	7.705	2.287	4.194	7.353	13.073
beta3_slope[11]	43.465	0.133	43.225	43.460	43.728
beta3_slope[12]	43.346	0.277	42.850	43.309	43.921
beta3_slope[13]	43.457	0.379	42.933	43.403	44.055
beta3_slope[14]	43.274	0.141	43.092	43.241	43.632
beta3_slope[15]	43.494	0.164	43.188	43.491	43.803
beta3_slope[16]	43.371	0.142	43.150	43.347	43.692
beta4_slope[11]	-0.730	0.166	-1.060	-0.731	-0.403
beta4_slope[12]	-1.160	0.455	-2.158	-1.127	-0.383
beta4_slope[13]	0.085	0.163	-0.236	0.087	0.399
beta4_slope[14]	-0.090	0.199	-0.471	-0.093	0.308
beta4_slope[15]	-0.767	0.161	-1.088	-0.766	-0.454
beta4_slope[16]	-0.159	0.176	-0.494	-0.160	0.192
sigma_H[1]	0.199	0.054	0.100	0.197	0.312
sigma_H[2]	0.171	0.030	0.119	0.169	0.236
sigma_H[3]	0.195	0.043	0.118	0.192	0.285
sigma_H[4]	0.423	0.079	0.294	0.413	0.597
sigma_H[5]	1.002	0.207	0.635	0.987	1.439
sigma_H[6]	0.382	0.199	0.029	0.376	0.781
sigma_H[7]	0.308	0.063	0.207	0.299	0.457
sigma_H[8]	0.415	0.088	0.273	0.405	0.606
sigma_H[9]	0.525	0.124	0.333	0.507	0.824
sigma_H[10]	0.217	0.043	0.144	0.213	0.312
sigma_H[11]	0.277	0.045	0.203	0.273	0.381
sigma_H[12]	0.438	0.164	0.207	0.417	0.767
sigma_H[13]	0.214	0.038	0.148	0.211	0.296
sigma_H[14]	0.508	0.094	0.349	0.501	0.714
sigma_H[15]	0.246	0.039	0.177	0.244	0.331
sigma_H[16]	0.224	0.044	0.153	0.219	0.325
lambda_H[1]	3.126	4.169	0.165	1.760	13.818
lambda_H[2]	8.175	7.640	0.745	6.044	27.368
lambda_H[3]	6.230	9.733	0.277	3.106	31.416
lambda_H[4]	0.006	0.004	0.001	0.005	0.017
lambda_H[5]	3.809	8.475	0.036	1.022	28.039
lambda_H[6]	7.010	14.058	0.008	0.691	42.908
lambda_H[7]	0.013	0.009	0.002	0.011	0.036
lambda_H[8]	8.299	10.037	0.117	4.744	37.355
lambda_H[9]	0.015	0.010	0.003	0.013	0.040
lambda_H[10]	0.295	0.404	0.035	0.197	1.060
lambda_H[11]	0.271	0.440	0.011	0.127	1.259
lambda_H[12]	4.986	6.681	0.182	2.805	23.986
lambda_H[13]	3.569	3.234	0.216	2.695	12.236
lambda_H[14]	3.353	4.190	0.230	1.993	14.809
lambda_H[15]	0.025	0.067	0.003	0.017	0.096
lambda_H[16]	0.834	1.297	0.038	0.438	3.932
mu_lambda_H[1]	4.316	1.860	1.237	4.133	8.307
mu_lambda_H[2]	3.828	1.917	0.653	3.665	7.827
mu_lambda_H[3]	3.519	1.885	0.836	3.187	7.870
sigma_lambda_H[1]	8.560	4.234	2.182	7.883	18.101
sigma_lambda_H[2]	8.361	4.609	1.058	7.747	18.457
sigma_lambda_H[3]	6.308	4.021	1.035	5.418	16.331
beta_H[1,1]	6.898	1.064	4.366	7.067	8.520
beta_H[2,1]	9.889	0.482	8.862	9.904	10.802
beta_H[3,1]	8.009	0.772	6.189	8.103	9.252
beta_H[4,1]	9.314	8.022	-7.058	9.511	24.682
beta_H[5,1]	0.105	2.229	-4.684	0.301	4.022
beta_H[6,1]	2.920	4.160	-7.212	4.478	7.631
beta_H[7,1]	0.565	5.928	-12.113	0.939	11.235
beta_H[8,1]	1.379	4.004	-2.332	1.220	3.532
beta_H[9,1]	12.820	5.701	1.138	12.779	24.432
beta_H[10,1]	7.021	1.726	3.354	7.105	10.338
beta_H[11,1]	5.057	3.571	-3.095	5.773	9.990
beta_H[12,1]	2.597	1.107	0.706	2.523	4.977
beta_H[13,1]	9.032	0.941	7.006	9.115	10.470
beta_H[14,1]	2.183	1.045	0.077	2.186	4.198
beta_H[15,1]	-6.130	3.861	-13.157	-6.349	2.108
beta_H[16,1]	3.461	2.704	-0.896	3.071	9.749
beta_H[1,2]	7.906	0.241	7.417	7.907	8.363
beta_H[2,2]	10.025	0.136	9.748	10.027	10.290
beta_H[3,2]	8.951	0.201	8.560	8.955	9.352
beta_H[4,2]	3.588	1.528	0.672	3.592	6.698
beta_H[5,2]	1.942	0.924	0.133	1.940	3.723
beta_H[6,2]	5.689	1.093	3.122	5.869	7.349
beta_H[7,2]	2.652	1.126	0.630	2.591	5.018
beta_H[8,2]	3.002	1.124	1.466	3.143	4.239
beta_H[9,2]	3.512	1.131	1.319	3.490	5.803
beta_H[10,2]	8.213	0.342	7.534	8.216	8.879
beta_H[11,2]	9.791	0.639	8.854	9.674	11.253
beta_H[12,2]	3.947	0.381	3.244	3.939	4.697
beta_H[13,2]	9.122	0.260	8.653	9.106	9.689
beta_H[14,2]	4.020	0.352	3.356	4.020	4.715
beta_H[15,2]	11.363	0.684	9.935	11.389	12.659
beta_H[16,2]	4.539	0.809	3.040	4.544	6.203
beta_H[1,3]	8.454	0.237	8.022	8.443	8.937
beta_H[2,3]	10.065	0.119	9.838	10.061	10.310
beta_H[3,3]	9.613	0.167	9.287	9.612	9.944
beta_H[4,3]	-2.512	0.878	-4.231	-2.512	-0.851
beta_H[5,3]	3.827	0.595	2.614	3.834	4.982
beta_H[6,3]	8.062	1.204	6.388	7.704	10.603
beta_H[7,3]	-2.774	0.673	-4.135	-2.754	-1.479
beta_H[8,3]	5.242	0.502	4.682	5.178	6.146
beta_H[9,3]	-2.845	0.740	-4.315	-2.837	-1.462
beta_H[10,3]	8.680	0.270	8.146	8.681	9.215
beta_H[11,3]	8.529	0.290	7.891	8.554	9.024
beta_H[12,3]	5.256	0.322	4.502	5.301	5.770
beta_H[13,3]	8.835	0.177	8.465	8.836	9.175
beta_H[14,3]	5.721	0.269	5.131	5.740	6.189
beta_H[15,3]	10.367	0.317	9.740	10.364	11.001
beta_H[16,3]	6.238	0.604	4.876	6.305	7.215
beta_H[1,4]	8.262	0.174	7.875	8.272	8.574
beta_H[2,4]	10.129	0.120	9.878	10.134	10.346
beta_H[3,4]	10.114	0.161	9.759	10.127	10.396
beta_H[4,4]	11.796	0.463	10.870	11.795	12.700
beta_H[5,4]	5.475	0.734	4.296	5.388	7.133
beta_H[6,4]	7.009	0.964	4.844	7.295	8.320
beta_H[7,4]	8.281	0.354	7.567	8.288	8.965
beta_H[8,4]	6.704	0.258	6.226	6.718	7.117
beta_H[9,4]	7.210	0.472	6.302	7.207	8.154
beta_H[10,4]	7.758	0.229	7.326	7.753	8.208
beta_H[11,4]	9.391	0.197	9.009	9.384	9.785
beta_H[12,4]	7.142	0.213	6.750	7.135	7.587
beta_H[13,4]	9.044	0.144	8.758	9.042	9.325
beta_H[14,4]	7.730	0.217	7.302	7.731	8.162
beta_H[15,4]	9.465	0.233	9.021	9.466	9.916
beta_H[16,4]	9.348	0.240	8.924	9.339	9.834
beta_H[1,5]	8.987	0.143	8.692	8.995	9.254
beta_H[2,5]	10.783	0.094	10.602	10.780	10.974
beta_H[3,5]	10.920	0.169	10.614	10.911	11.277
beta_H[4,5]	8.391	0.477	7.484	8.390	9.331
beta_H[5,5]	5.430	0.570	4.117	5.480	6.481
beta_H[6,5]	8.836	0.657	7.897	8.671	10.355
beta_H[7,5]	6.757	0.348	6.108	6.751	7.477
beta_H[8,5]	8.217	0.227	7.855	8.203	8.651
beta_H[9,5]	8.210	0.482	7.241	8.207	9.149
beta_H[10,5]	10.083	0.226	9.639	10.081	10.520
beta_H[11,5]	11.512	0.223	11.073	11.515	11.953
beta_H[12,5]	8.491	0.199	8.100	8.492	8.906
beta_H[13,5]	10.003	0.131	9.745	10.004	10.253
beta_H[14,5]	9.209	0.236	8.764	9.198	9.698
beta_H[15,5]	11.176	0.249	10.671	11.182	11.652
beta_H[16,5]	9.906	0.176	9.547	9.912	10.232
beta_H[1,6]	10.179	0.185	9.852	10.167	10.585
beta_H[2,6]	11.513	0.107	11.307	11.512	11.731
beta_H[3,6]	10.815	0.157	10.475	10.824	11.104
beta_H[4,6]	12.883	0.839	11.140	12.912	14.541
beta_H[5,6]	5.893	0.588	4.797	5.876	7.118
beta_H[6,6]	8.720	0.710	6.896	8.873	9.716
beta_H[7,6]	9.876	0.578	8.714	9.876	10.998
beta_H[8,6]	9.513	0.302	9.019	9.532	9.969
beta_H[9,6]	8.455	0.805	6.898	8.441	10.088
beta_H[10,6]	9.516	0.308	8.862	9.541	10.046
beta_H[11,6]	10.808	0.345	10.077	10.842	11.422
beta_H[12,6]	9.377	0.256	8.891	9.365	9.900
beta_H[13,6]	11.047	0.162	10.754	11.037	11.378
beta_H[14,6]	9.817	0.294	9.231	9.816	10.394
beta_H[15,6]	10.820	0.437	9.941	10.824	11.699
beta_H[16,6]	10.536	0.239	10.012	10.544	10.980
beta_H[1,7]	10.888	0.846	8.810	10.997	12.304
beta_H[2,7]	12.200	0.436	11.272	12.217	13.017
beta_H[3,7]	10.538	0.661	9.058	10.604	11.615
beta_H[4,7]	2.493	4.308	-5.790	2.369	11.441
beta_H[5,7]	6.441	1.748	3.082	6.400	10.112
beta_H[6,7]	9.737	2.523	4.864	9.590	16.321
beta_H[7,7]	10.453	2.878	4.940	10.463	16.215
beta_H[8,7]	10.956	1.139	9.339	10.901	12.780
beta_H[9,7]	4.458	4.027	-3.560	4.511	12.279
beta_H[10,7]	9.789	1.409	7.190	9.735	12.763
beta_H[11,7]	11.004	1.720	7.857	10.898	14.714
beta_H[12,7]	10.012	0.972	7.835	10.095	11.617
beta_H[13,7]	11.667	0.758	9.842	11.762	12.816
beta_H[14,7]	10.403	0.954	8.365	10.459	12.121
beta_H[15,7]	12.147	2.271	7.777	12.093	16.816
beta_H[16,7]	12.310	1.288	10.192	12.132	15.293
beta0_H[1]	8.696	13.369	-18.517	9.009	34.788
beta0_H[2]	10.544	6.750	-2.898	10.536	23.897
beta0_H[3]	9.853	9.948	-9.781	9.805	29.589
beta0_H[4]	0.474	180.145	-363.343	0.783	357.572
beta0_H[5]	4.759	24.225	-42.143	4.605	54.439
beta0_H[6]	6.701	57.601	-110.790	7.701	123.849
beta0_H[7]	6.462	137.600	-272.340	5.282	292.186
beta0_H[8]	6.367	34.984	-16.304	6.380	28.445
beta0_H[9]	3.974	119.653	-238.320	3.462	248.371
beta0_H[10]	8.886	32.717	-56.587	8.537	76.787
beta0_H[11]	8.542	50.300	-105.418	9.254	114.101
beta0_H[12]	6.452	11.687	-15.981	6.678	28.622
beta0_H[13]	9.713	10.471	-11.352	9.533	31.222
beta0_H[14]	7.214	11.667	-16.343	7.071	29.667
beta0_H[15]	9.658	112.396	-210.524	9.555	249.076
beta0_H[16]	8.590	26.048	-43.013	8.291	61.526